Probability question. 52 card deck

[peteboymoron]

hi i have a question for hw. i can't seem to get the ans

you are going to draw 4 times, with replacement, from a deck of 52 cards. what is the probability that you will get 2 aces ( you drawing 4 four cards )

the ans is 0.030251
but i can't seem to be able to get to the problem. please work out the solution, thanks a lot!


peter

 

[cookiemonster]

P(drawing an ace) = 4/52 = 1/13
P(not drawing an ace) = 1 - 1/13 = 12/13

Each drawing is independent, so we multiply probabilities. We want two aces and two non-aces, so,

(1/13)(1/13)(12/13)(12/13)

However, this is the probability for drawing aces the first two cards and then non-aces the second two. We could also draw two non-aces and then draw two aces, or one non-ace, then two aces, then another non-ace, etc. There are, in fact, $\binom{4}{2}$ different orders to draw the aces. Each one has the same probability, so our total probability becomes

$$\binom{4}{2}\frac{1}{13}\frac{1}{13}\frac{12}{13}\frac{12}{13} = .030251$$.

cookiemonster

 

[peteboymoron]

*here a cookie

thanks thanks!

 

[rattis]

This is a binomial question. English people will probably be more familar with writing as 4C2 on a calculator(nCr button) (4 ways out of 2).

 

[matt grime]

Writing nCr is useful in html, but the de facto universal symbol is the one used above using braces. It's only in secondary education (UK, don't know about others, but nCr seems widely used in the US too from what I recall) that some people insist on using the nCr notation and don't tell their students there is a more widely used one. Just as they insist on calling it the Argand plane/diagram, when no mathematician (professional) would use that terminology in an article.And I am an English mathematician. And one who got flamed for requesting someone not just tell me they were doing sophomore calc when attempting to describe the level of the course they were doing.

 

[cookiemonster]

I'm in the States and I originally learned the nCr notation back in second year algebra or something because the textbook used nCr notation. We were then shown the notation I used above in pre-calculus when discussing the binomial theorem because the textbook used that notation. The textbooks we used are, if I recall correctly, hugely popular in US schools, so I'd say that most students here are familiar with both by the time they get out of high school.

I was never a huge fan of the nCr notation, but I always thought the braces notation looked pretty tidy, so that's what I use.

cookiemonster

 

[rattis]

nCr notation is easier to lay out

 

[jeromedjames]

Hello I have a question similar to Peter's

If one card is selected at random from a deck of cards. What is the probability that the card selected is a red card or a picture card.

 

[Nimz]

All you have to do for that one is count how many cards are red or picture cards and divide by fifty-two.

Alternatively, count the red cards, count the picture cards, and count the red picture cards. If you add the first two numbers together, you will have counted the red picture cards twice, so subtract that number. That'll be the same total as the first method.

It's good to be familiar with all the useful notation. I prefer the binomial notation of $\binom{n}{r}$, but _nC_r and C(n;r) have their niches. The latter, e.g. when extending to C(n;r₁,r₂,...,r_m).